Question

A 4-bit message has been encoded with the Hamming code H(7,4) and transmitted over a possibly noisy channel with at most one error. The message 0100101 (binary) is received.

Hi,

I found an error at Parity 6 and the original 4 bit message is 0100111. I was told that I was wrong. Can someone help and explain why?

Thanks

Answer 1

There are only three parity bits in H(7,4); those bits are at (one-indexed) positions 1, 2 and 4. There is no ‘parity 6’ to check. Let’s examine the received message:

1. Parity bit 1 at position 1 covers bits 1, 3, 5 and 7. Those bits are 0, 0, 1 and 1, respectively. We take the sum of these bits, which comes to 2. This is an even sum, so we assume this bit is safe.

2. Parity bit 2 at position 2 covers bits 2, 3, 6 and 7. Those bits are 1, 0, 0 and 1, respectively. Again, the sum of these bits is even, so no problem exists yet.

3. Parity bit 3 at position 4 covers bits 4, 5, 6 and 7. Those bits are 0, 1, 0 and 1, respectively. The sum is even, so no problem here either.

The parity checks all add up, so there’s no indication of error in the received message.